Dominated Convergence Theorem
The Dominated Convergence Theorem is a fundamental result in measure theory and integration. It provides conditions under which the limit of a sequence of functions can be interchanged with the integral. Specifically, if a sequence of measurable functions converges pointwise to a function and is dominated by an integrable function, then the integral of the limit equals the limit of the integrals.
This theorem is particularly useful in probability theory and real analysis, as it allows for the simplification of complex integrals. It ensures that under certain conditions, we can safely evaluate limits and integrals without losing accuracy in the results.